Fuzzy Entropy

Three things change; the entropy logic does not.

Cells. In crisp entropy every cell is a single number. Here every cell is three numbers: lowest, most likely, highest. No weight is entered, because this method already generates weights; the input is only the decision matrix and, for each criterion, its direction (more or less is better).

Scale equalisation. Crisp entropy converts cost criteria by dividing the smallest value by every value, then converts the column to a probability by dividing by the column sum. This extension uses a different normalisation: for each component (lowest, most likely, highest), it computes (value minus the column minimum) for a benefit criterion, or (the column maximum minus the value) for a cost criterion, and then turns this difference into a share by dividing by the column sum. This is a shift-and-sum normalisation, "taking the difference from the smallest", rather than the "dividing by the smallest" of the crisp ENTROPY card; the two methods can give different probability distributions, and hence different entropy values, on the same data.

Entropy and weight. On each component, that is, the lowest, most likely and highest matrices, an independent crisp entropy calculation is run with the normalisation above: conversion to a probability, Shannon entropy, divergence (1 minus entropy), and a weight that sums to 1. Three separate weight vectors emerge from the three components; these three vectors are averaged and renormalised so that the sum is again 1, giving the final weight. In other words, the fuzziness is split across three parallel crisp calculations and only "settles" through averaging at the end; no triangular arithmetic, addition or multiplication, is used at any step.

DecisionMind fixes three things in this extension: the shift-and-sum normalisation (benefit: value minus minimum, cost: maximum minus value), the independent crisp entropy calculation for each of the three components, and the rule that the final weight is the plain arithmetic mean of the three component weights, not their centre of gravity.

The output, as in crisp entropy, is a weight vector summing to 1, and it is read on the same principle: a weight shows not how important the decision-maker considers the criterion, but how much this criterion discriminates between the alternatives in this table.

The difference is this. The further apart the weights given by the three components (lowest, most likely, highest) sit from one another, the wider the triangle, and the more uncertain that criterion's weight becomes. If the three component weights sit close together, that is, if the triangle's width barely affects the weight, the result is robust. Where the components diverge sharply, a single average number conceals that uncertainty.

Thus instead of writing:

"The fuzzy entropy weight came out at 0.358, so this criterion determines close to a third of the decision"

the report should read:

"The average weight is 0.358; whether the weights given separately by the lowest, most likely and highest components sit close together, or far apart, shows how robust this result is, and it should be stated in the report"

Use this extension when the values in the decision table are not exact measurements but expert estimates or approximate magnitudes; the rule given on the fuzzy data-type card applies here in exactly the same way. Crisp entropy's exit condition also applies here: every value must be positive, and if a criterion is identical across all alternatives its weight comes out at zero and that criterion should be removed from the table. In addition, every triangle must satisfy lowest ≤ most likely ≤ highest; no criterion may be constant across all three components at once, otherwise that criterion's weight becomes undefined in all three components.

Uniform proportional fuzzification. Widening every cell by the same proportion, say ±10 per cent, to build the triangle makes the three components' normalised weights nearly identical to one another; in that case the fuzzy extension adds no information over the crisp method, and this is a situation the manifest itself warns about.

Assuming there is one natural, canonical "fuzzy entropy" formula. The literature contains more than one formulation of fuzzy entropy weighting, with different normalisations and different forms of fuzzy arithmetic; the independent-calculation-then-average approach DecisionMind uses here is only one of them, not the single "correct" method.

Confusing this scale equalisation with the crisp ENTROPY card's. This extension uses "taking the difference from the smallest" normalisation, not "dividing by the smallest"; the two methods can give different weights on the same table, and which one was used should be stated in the report.

Over-trusting a weight produced from few alternatives. The warning on the crisp entropy card holds even more strongly here: a change at just one end of a fuzzy cell, for instance in its highest value, can shift the weight noticeably in a table built from a small number of alternatives.

The governing principle is this:

The fuzzy entropy weight is built by computing the three components, lowest, most likely and highest, separately and then averaging them; the larger the gap between the components, the more fragile the average weight is, and this fragility should be shown in the report.